Mathematical Analysis, Second Edition

1 PREFACEA glance at the table of contents will reveal that this textbooktreats topics inanalysis at the "Advanced Calculus" level. The aim has beento provide a develop-ment of the subject which is honest, rigorous, up to date, and, at thesame time,not too book provides a transition from elementary calculustoadvanced courses in real and complex function theory, and it introducesthe readerto some of the abstract thinking that pervades modern Second Edition differs from the first inmany respects. Point set topologyis developed in the setting of general metricspaces as well as in Euclidean n-space,and two new chapters have been addedon Lebesgue integration. The material online integrals, vector analysis , and surface integrals has beendeleted. The order ofsome chapters has been rearranged, many sections have been completely rewritten,and several new exercises have been development of Lebesgue integration follows the Riesz-Nagyapproachwhich focuses directly on functions and their integrals and doesnot depend onmeasure treatment here is simplified, spread out, and somewhatrearranged for presentation at the undergraduate first Edition has been used in mathematicscourses at a variety of levels,from first-year undergraduate to first-year graduate, bothas a text and as supple-mentary Second Edition preserves this example,Chapters 1 through 5, 12, and 13 providea course in differential calculus of func-tions of one or more variables.

2 Chapters 6 through 11, 14, and15 provide a coursein integration theory. Many other combinationsare possible; individual instructorscan choose topics to suit their needs by consulting the diagram on the nextpage,which displays the logical interdependence of the would like to express my gratitude to themany people who have taken thetrouble to write me about the first comments and suggestionsinfluenced the preparation of the Second thanks are due Aliprantis who carefully read the entire manuscriptand madenumerous helpful also provided some of the new , I would like to acknowledge my debt to the undergraduatestudents ofCaltech whose enthusiasm for mathematics provided the original incentivefor 1973 LOGICAL INTERDEPENDENCE OF THE CHAPTERS1 THE REAL AND COM-PLEX NUMBER SYSTEMS2 SOME BASIC NOTIONSOF SET THEORY3 ELEMENTS OF POINTSET TOPOLOGYI4 LIMITS ANDCONTINUITYI5 DERIVATIVESI6 FUNCTIONS OF BOUNDEDVARIATION AND REC-TIFIABLE CURVES8 INFINITE SERIES ANDINFINITE PRODUCTS7 THE RIEMANN-STIELTJES INTEGRAL12 MULTIVARIABLE DIF-FERENTIAL CALCULUS9 SEQUENCES OFFUNCTIONS10 THE LEBESGUEINTEGRALI13 IMPLICIT FUNCTIONSAND EXTREMUMPROBLEMS14 MULTIPLE RIEMANNINTEGRALS11 FOURIER SERIES ANDFOURIER INTEGRALS1615 CAUCHY'S THEOREM ANDMULTIPLE LEBESGUETHE RESIDUE CALCULUSINTEGRALSCONTENTSC hapter 1 The Real and Complex Number.

3 Field .. order .. representation of real .. unique factorization theorem for .. numbers.. numbers.. bounds, maximum element, least upper bound(supremum).. completeness .. Some properties of the supremum.. of the integers deduced from the completeness The Archimedean property of the real-number . numbers with finite decimal . decimal approximations to real numbers.. decimal representation of real .. values and the triangle .. Cauchy-Schwarz inequality.. and minus infinity and the extended real number system R* .. representation of complex numbers.. imaginary unit.. value of a complex .. of ordering the complex numbers.. properties. of complex exponentials.. argument of a complex .. powers and roots of complex .. logarithms.. sines and .. and the extended complex plane C*..24 Exercises..25viContentsChapter 2 Some Basic Notions of Set .. pairs.. product of two sets.. and .. terminology concerning.

4 Functions and .. (equinumerous) sets.. and infinite sets.. and uncountable .. of the real-number .. algebra.. collections of countable sets..42 Exercises..43 Chapter 3 Elements of Point Set .. space R".. balls and open sets in R".. The structure of open sets in .. points. Accumulation .. sets and adherent points.. Bolzano-Weierstrass theorem.. Cantor intersection .. The LindelSf covering theorem.. Heine-Borel covering theorem.. in R".. set topology in metric spaces.. subsets of a metric space.. of a ..65 Chapter 4 Limits and .. sequences in a metric .. metric spaces .. of a .. of complex-valued functions.. of vector-valued .. of composite .. complex-valued and vector-valued functions.. of continuous .. and inverse images of open or closed sets.. continuous on compact .. mappings (homeomorphisms).. 's .. of a metric .. continuity.. continuity and compact sets.. theorem for contractions.

5 Of real-valued ..94 Exercises..95 Chapter .. of .. and .. of .. chain .. derivatives and infinite .. with nonzero .. derivatives and local .. 's .. The Mean-Value Theorem for .. theorem for derivatives.. 's formula with .. of vector-valued .. of functions of a complex .. The Cauchy-Riemann ..121 Chapter 6 Functions of Bounded Variation and Rectifiable .. of monotonic functions.. of bounded variation.. variation.. property of total .. variation on [a, x] as a function of .. of bounded variation expressed as the difference ofincreasing functions.. functions of bounded .. and paths.. paths and arc .. and continuity properties of arc .. of of parameter..136 Exercises..137 Chapter 7 The Riemann-Stieltjes .. definition of the Riemann-Stieltjes integral.. by .. of variable in a Riemann-Stieltjes integral.. to a Riemann .. functions as integrators.. of a Riemann-Stieltjes integral to a finite.

6 's summation formula.. increasing and lower and linearity properties of upper and lower 's condition.. theorems.. of bounded variation.. conditions for existence of Riemann-Stieltjes conditions for existence of Riemann-Stieltjes Value Theorems for Riemann-Stieltjes .. The integral as a function of the .. fundamental theorem of integral calculus.. of variable in a Riemann integral.. Mean-Value Theorem for Riemann .. integrals depending on a parameter.. under the integral sign.. the order of .. 's criterion for existence of Riemann integrals.. Riemann-Stieltjes ..174 Chapter 8 Infinite Series and Infinite .. and divergent sequences of complex .. superior and limit inferior of a real-valued sequence.. sequences of real .. and removing parentheses.. series.. and conditional .. and imaginary parts of a complex for convergence of series with positive terms.. geometric .. integral .. big oh and little oh.

7 Ratio test and the root .. 's test and Abel's test.. sums of the geometric series Y. z" on the unit circle Iz1= 1.. of series.. 's theorem on conditionally convergent .. theorem for double .. A sufficient condition for equality of iterated of series.. 's product for the Riemann zeta function..209 Exercises..210 Chapter of FunctionsPointwise convergence of sequences of .. of sequences of real-valued functions.. of uniform convergence.. convergence and continuity.. Cauchy condition for uniform .. convergence of infinite series of .. A space-filling .. convergence and Riemann-Stieltjes integration.. convergent sequences that can be integrated term .. convergence and .. conditions for uniform convergence of a . convergence and double .. series.. of power .. The substitution .. of a power series.. power .. The Taylor's series generated by a function.. 's .. binomial .. 's limit .. 's ..246 Exercises.

8 247 Chapter Lebesgue IntegralIntroduction.. integral of a step .. sequences of step .. functions and their .. functions as examples of upper The class of Lebesgue-integrable functions on a general properties of the Lebesgue .. integration and sets of measure .. Levi monotone convergence .. The Lebesgue dominated convergence theorem.. of Lebesgue's dominated convergence theorem.. integrals on unbounded intervals as limits of integrals onbounded intervals.. Riemann integrals.. functions.. of functions defined by Lebesgue integrals.. under the integral sign.. the order of integration.. sets on the real .. Lebesgue integral over arbitrary subsets of .. integrals of complex-valued functions.. products and norms.. set L2(I) of square-integrable functions.. set L2(I) as a semimetric .. A convergence theorem for series of functions in L2(I).. Riesz-Fischer ..297 Exercises..298 Chapter 11 Fourier Series and Fourier.

9 Systems of .. theorem on best . Fourier series of a function relative to an orthonormal system .. of the Fourier coefficients.. Riesz-Fischer .. convergence and representation problems for trigonometric Riemann-Lebesgue lemma.. Dirichlet integrals.. An integral representation for the partial sums of a Fourier 's localization theorem.. conditions for convergence of a Fourier series ata .. summability of Fourier series.. of Fej6r's .. Weierstrass approximation .. forms of Fourier series.. Fourier integral .. exponential form of the Fourier integral .. transforms.. convolution theorem for Fourier .. Poisson summation ..335 Chapter 12 Multivariable Differential .. The directional derivative.. derivatives and .. The total .. total derivative expressed in terms of partial derivatives.. An application to complex-valued functions.. matrix of a linear function.. Jacobian matrix.. chain .. form of the chain rule.

10 Mean-Value Theorem for differentiable .. A sufficient condition for differentiability.. A sufficient condition for equality of mixed partial 's formula for functions from R" to RI..361 Exercises..362 Chapter 13 Implicit Functions and Extremum .. with nonzero Jacobian determinant.. inverse function theorem.. The implicit function theorem.. of real-valued functions of one variable.. of real-valued functions of several variables.. problems with side conditions..380 Exercises..384 Chapter 14 Multiple Riemann .. measure of a bounded interval in R".. Riemann integral of a bounded function defined on a compactinterval in R".. of measure zero and Lebesgue's criterion for existence of a multipleRiemann .. of a multiple integral by iterated integration.. sets in R".. integration over Jordan-measurable sets.. content expressed as a Riemann integral.. property of the Riemann .. Theorem for multiple ..402 Chapter 15 Multiple Lebesgue.